A352512 - OEIS

%I #10 Mar 31 2022 03:04:30

%S 0,1,0,1,0,0,2,1,0,0,1,0,1,2,1,1,0,0,1,0,0,1,0,0,1,1,2,2,2,1,1,1,0,0,

%T 1,0,0,1,0,0,0,0,1,1,1,0,0,0,1,1,1,1,3,2,2,2,1,2,1,1,1,1,1,1,0,0,1,0,

%U 0,1,0,0,0,0,1,1,1,0,0,0,0,0,0,0,2,1,1

%N Number of fixed points in the n-th composition in standard order.

%C The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. See also A000120, A059893, A070939, A114994, A225620.

%C A fixed point of composition c is an index i such that c_i = i.

%F A000120(n) = A352512(n) + A352513(n).

%e The 169th composition in standard order is (2,2,3,1), with fixed points {2,3}, so a(169) = 2.

%t stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];

%t Table[pq[stc[n]],{n,0,100}]

%Y The version counting permutations is A008290, unfixed A098825.

%Y The triangular version is A238349, first column A238351.

%Y Unfixed points are counted by A352513, triangle A352523, first A352520.

%Y A011782 counts compositions.

%Y A088902 gives the fixed points of A122111, counted by A000700.

%Y A352521 counts comps by strong nonexcedances, first A219282, stat A352514.

%Y A352522 counts comps by weak nonexcedances, first col A238874, stat A352515.

%Y A352524 counts comps by strong excedances, first col A008930, stat A352516.

%Y A352525 counts comps by weak excedances, first col A177510, stat A352517.

%Y Cf. A088218, A114088, A115994, A238352, A330644, A350841, A352486.

%K nonn

%O 0,7

%A _Gus Wiseman_, Mar 26 2022